This invention relates to a method and apparatus for measuring fluid density and for measuring parameters which may cause the density of the fluid to change.
It has been known for some time that a vibratory element, such as a quartz crystal, when exposed to a gas will change its frequency as the gas pressure changes. See, for example, M. A. Cotter, U.S. Pat. No. 4,126,049, J. W. Stansfeld, U.S. Pat. No. 4,232,544, P. N. Potter, U.S. Pat. No. 4,178,804, "Vacuum Microbalance Techniques", Edited by Klaus H. Behrndt, 1966, Plenum Press, N.Y., and A. Genis and D. E. Newell, "Using The X-Y Flexure Watch Crystal as a Pressure-Force Transducer" delivered at the Thirty First Annual Frequency Control Symposium, Atlantic City, N.J., June, 1977. Although these references recognize that the frequency of a vibratory element will change with a change in pressure of the gas surrounding the element, none recognized or disclosed that gas density measurement could be utilized as the mechanism for measuring pressure, and thus none recognized the full potential of gas density determination as a vehicle for measuring a variety of other parameters such as temperature, acceleration, flow, force, differential pressure, etc. One prior art reference, Theory of Vibrating Systems and Sound, by Irving B. Crandell, Von Nostrand, 1926, pp. 124-133, disclosed that the frequency of a vibrating string or rod would change with a change in density of the medium in which the string was placed, but failed to recognize how this phenomenon could be employed in practice and, in fact, misleadingly stated that the change in frequency resulting from a change in density would not be great unless the density of the medium were comparable to that of the rod.
It has been discovered that a principal mechanism by which a flexure or torsional mode vibratory element exposed to a fluid (liquid or gas) is caused to change its frequency is not an intrinsic change in pressure but rather is a change in the density of the fluid. The effect is equivalent to that of adding mass to the vibratory element (increase in density) or taking mass from the vibratory element (decrease in density) to respectively reduce or increase the frequency of vibration. Viewed another way, the effect may be likened to that of an object "flapping in the wind". It has been found that when the surface area of the object normal to the direction of movement or vibration of the object is increased, the "pushing" of the fluid by the object becomes more difficult resulting in a slowing of the vibration, and vice versa. Recognizing this mechanism allows for optimizing the sensitivity of vibratory devices used for measuring density directly or for measuring other parameters such as pressure, temperature, etc. In particular, the greater the surface area of the vibratory element, the greater will be the sensitivity of the frequency of vibration to fluid density changes. Also, by proper selection of the working fluid in which the vibratory element is placed, different measurement objectives can be achieved. For example, in a gas density device, use of a more dense gas such as argon will give rise to greater sensitivity, whereas use of a less dense gas, such as helium, will allow detection of density (and thus pressure or other parameter) changes over a wider range.
The above will become more apparent from a mathematical analysis of the effect fluid density has on a vibratory element. The pressure due to the frontal area of a rectangular bar having a thickness T and a width w, which is vibrating in the w direction, moving ambient fluid is given by: EQU P=-2.pi.frU.sub.0 TC sin (2.pi. ft), (1)
where f is the frequency of vibration, r is the density of the ambient fluid, U.sub.0 is the peak vibration velocity of the bar in the w direction, and C is a shape factor for edge effects and has a value near unity. The force F per unit length acting on the bar due to the pressure is 2PT, since both sides of the bar are acting on the fluid. Therefore, EQU F=-4.pi.frU.sub.0 T.sup.2 C sin (2.pi. ft). (2)
Since the velocity U of the vibrating bar is related to the displacement Y=Y.sub.0 sin (2.pi. ft) by: EQU U=2.pi.fY.sub.0 cos (2.pi. ft), (3)
then the force F per unit length is given by: EQU F=-8.pi..sup.2 f.sup.2 rY.sub.0 T.sup.2 C sin (2.pi. ft). (4)
The conventional equation for equilibrium of a laterally vibratintg beam with the force F, from equation 4, added to the inertial forces is: ##EQU1## where E, I, and r.sub.q are Young's Modulus, moment of inertia, and density of the vibrating beam respectively. The resonant frequency f.sub.0, for r=0 (no ambient fluid), is thus perturbed by the added term 2rT.sup.2 C in equation 5. The added term is of the form of an added mass per unit length, just as r.sub.q Tw is the mass per unit length of the bar. The resulting resonant frequency f, from equation 5, is given by ##EQU2## For small perturbations, ##EQU3## From the last equation, it is apparent that the frequency of a vibrating element is substantially linearly dependent upon the density of the working fluid and that increased sensitivity of a vibratory element can be achieved by both increasing the thickness-to-width ratio of the vibrating bar and selecting a more dense fluid as the working medium.